discrete-time multivariable adaptive control

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Discrete Time Multivariable

daptive ontrol

GRAHAM C. GOODWIN, PETER J. RAMADGE, AND PETER E. CAINES

ADAPTIVE

control, so great is its appeal, has been stud

ied for almost forty years. Its early history is one of diverse,

but interesting, heuristic endeavors. One of the first problems

studied, and one that has formed the focus of much subsequent

research, is that ofadaptivemodel reference control inwhich the

controller employs an estimate of the unknown plant to cause

the output y to track y*

= H r,

where

r

is the reference,

H

a

prespecified reference model, and y* the desired output. An

other consistent theme is certaintyequivalence, regarding (and

using) the estimated plant as if it were the true plant; the moti

vation is obvious . . . if the estimated parameters converge to the

true parameters, the certainty equivalent controller converges to

a satisfactory (optimal) controller for the true plant. However,

adaptive controllers can perform satisfactorily even if, as is of

ten the case, the estimated parameters do not converge to their

true values; this was dramatically shown in the paper by Astrom

and Wittenmark [1], also included in this volume. Early use of

Lyapunov theory to establish stability was restricted to mini

mum phase plants with relative degree one. Plants with a rela

tive degree higher than one posed a major problem [17] that was

not satisfactorily resolved until the late 1970s; indeed, the paper

by Goodwin, Ramadge, and Caines was a major contribution

to the resolution of this formidable problem and a substantial

stimulus to subsequent research.

Suppose the system to be controlled is described by

Ay

=Bu

where A and B are polynomials of degree nandm, respectively,

in the shift operator

q,

and

A

is monic; for continuous-time sys

tems q is replaced by the derivative operator

d

/

d t.

The system

may be re-parameterized (as shown in [1] and [15], the latter in

'state-space language') to obtain

Ey = Ry + Su

where E = E

1E

2

is monic and Hurwitz, and

£1

and E

2

have

degrees nand m, respectively. Hence (ignoring exponentially

decaying terms)

where

YE

:= (1 /

E)y, UE :=

(1 /E)u,

()

is a vector of the coeffi

cients of the polynomials Rand S, and the regressorvector ¢ has

components q i

y E,

i

= 0,

.. .

,

n

- 1, and q i

U

E,

i

= 0,

.. .

,

n.

The estimation equation

y

=

¢T

e

may be used in many differ

ent ways to provide an estimate

e

of the unknown parameter

e.

Supposethe desired closed-loop transferfunction is H = (N / E)

so that the desired output is

y* =

Hr , and that b.;

=

Sn

=

1 is

known. To determine the control, the system equation may be

written in the form

E2Y = RYE

l

+ SUEl = 1/JTe

where 1/J:=E

2

¢

(the vector 1/1 has componentsqiYE

l

, i

=

0, .. . ,

n - 1, and

qiUEl ' i

= 0,

. . .

, n). The degree of

E

1

is n so that

1/JT emay be written as

1JT

e+ u. If eisknown (and the system is

minimum phase), the control u may be chosen (as u

=

E2Y*

1JT

()) to cause the output to satisfy

E2Y =

1JT

e

+ U

=

E2Y*

so that the tracking error

y - y*

decays to zero (and all signals

remain bounded). If () is unknown, a tempting strategy is to

employ the certainty equivalent control u =

E2Y* -

1JTewith

the result that E

2

y* = 1/JT e and the output is now given by

E2Y

= E

2

y* + t/JTe

The tracking error er := Y - y* satisfies

e t =

e

+

er

where y:= ¢Teis the estimated output, e :=

Y -

Y

=

¢Teis

the prediction error (Monopoli's augmented error),

e

:=

() -

e

is the parameter error, and er :=

y -

y*) is an estimate of the

tracking error

eT.

Since

eT =

[(1/E2)l/ T]e -

(1/£2)[

l/IT OJ

it is also known as the 'swapping error'; eT is zero if eis con

stant, but otherwise depends on the rate of change of e(in the

continuous-time case it is proportional to (d / dt)e).

Monopoli [15] proposed use of the certainty equivalent con

trollaw inconjuctionwith a simple gradientestimator

(d/ dt)O

=

¢e; however, as was later pointed out, convergence of the

85



tracking error to zero was not established except for the case

of unity relative degree. The first proof of convergence (for

continuous-time systems) was achieved in [4] by modifying the

controller; nonlinear dampingwas added to ensure convergence

of eT to zero; this enabled convergence of

er

(and bounded

ness of all signals) to be proven. Because

of

the complexity

of

the controller, and of the accompanying analysis, the paper [4]

did not have the impact it deserved. The second and much sim

pler proof (for discrete-time, systems), presented in this paper

by Goodwin, Ramadge, and Caines, achieved convergence by

modifying the

estimator;

the rate of change of

8

was reduced

(thereby reducing eT)by the introduction of error normalization

(replacing

e

in the estimator algorithm

bye := e/[I, + 1c/J1

2

]1/2).

Error normalization was independently proposed in [3].

The relevant result is Lemma 3.1 of the paper by Goodwin,

Ramadge, and Caines, which essentially states:

(a) Suppose the

estimator

is such that the normalizedpredic

tion errore

=

e/[1

+

1c/J1

2

]1/2 lies in

f

2

•

(b) Suppose the

controller

is such that the regressorvector c/J

satisfies the growth condition

1c/J(t)1

2

c[1+ max{le(r)1

2

I rE

[0, t]}

for all t o.

Then:

(i)

Ic/J(t)

I is uniformlybounded, and

(ii)

e(t) 0 as t

00.

The result is simple to state and prove. It provideda simple means

for establishing convergence (and boundedness of all signals) for

a wide range of adaptive controllers and contributed to an explo

sive re-awakening of interest in adaptive control.

An

important

feature of the result is itsmodularity: condition (a) on the estima

tor can be established independently of (b), i.e., independently

of

the control law. The result was presented in an electrifying

seminar at Yale University (New Haven, Connecticut) in late

1978 and was rapidly extended to the deterministic continuous

time case in [16] and [21]. All three papers appeared in the same

issue of the IEEETransactionsonAutomaticControl; ironically,

the paper by Goodwin, Ramadge, and Caines was the only one

that did not appear as a regular paper. A parameter estimation

perspective of the continuous-time results was given in

[5],

pro

viding an analogous modular decomposition of the conditions

for convergence. Stochastic versions of the paper by Goodwin,

Ramadge, and Caines and of

[5]

appeared, respectively, in

[7]

and [6]. Research on convergence and stability has continued to

this day. For example, the earlier nonlinear damping approach

of [4] was extended in [11] to deal with 'true' output nonlin

earities for which certainty equivalent adaptive controllers are

inadequate,andbackstepping

[13]

was developedas a systematic

tool for designing adaptive controllers for linear and nonlinear

systems with high relative degree. Furthermore, switching was

introduced to enforce convergence when structural properties

(e.g., relative degree) are unknown

[18],

and modifications were

made in the basic algorithm to ensure robustness

[19], [9], [12].

Researchresults of this period were rapidly consolidated in texts

such as [8], [20], [14], [2], [22], and [10].

The paper by Goodwin, Ramadge and Caines is an important

milestone in the evolution of adaptive control. It contributed

much to the richness of a subject that has progressed far and that

now appears poised for further significant advances.

REFERENCES

[1] K.1.

ASTROM AND

B.

WITTENMARK,

On self tuning regulators,

Auto

matica,

9:185-199, 1973.

[2] K.1.

ASTROM AND

B.

WITTENMARK,AdaptiveControl,

Addison-Wesley

(Reading,MA), 1989.

[3] B.

EGARDT, StabilityofAdaptiveControllers,

Springer-Verlag(NewYork),

1979.

[4] A. FEUER AND S. MORSE, Adaptivecontrolof single input, single output

linear systems,

IEEE

Trans.

Automat.

Contr., AC-23(4):557-569, 1978.

[5]

G.C. GOODWIN AND D.

Q. MAYNE, A

parameterestimationperspective

.of continuous timeadaptivecontrol, Automatica,23:57-70,1987.

[6] G.C.GOODWINAND D.Q. MAYNE, Continuous-timestochasticmodelref

erenceadaptivecontrol,

IEEE Trans. Automat.Contr.,

AC-36(ll):

1254

1263, November1991.

[7]

G.C. GOODWIN,P.J. RAMADGE, AND P.

E. CAINES,

Discretetime stochas

tic adaptivecontrol, SIAMJ. Contr. Optimiz.,19(6):829-853,1981.

[8]

G.C. GOODWIN AND K.S.

SIN,AdaptiveFiltering,Predictionand

Control,

PrenticeHall (EnglewoodCliffs,NJ), 1984.

[9] P.A. IOANNOU AND P.KOKOTOVIC,

AdaptiveSystemswithreducedmodels,

Lecture Notes in Control and Information Sciences,

Vol.

47, Springer

Verlag(NewYork),1983.

[10] P. A. IOANNOU

AND

1. SUN,

Robust Adaptive Control,

Prentice Hall

(EnglewoodCliffs,NJ), 1989.

[11] I. KANELLAKOPOULOS, P. V.

KOKOTOVIC, AND

A. S. MORSE, Adap

tiveoutput-feedbackcontrol of systemswithoutput nonlinearities, IEEE

Trans.

Automat.Contr., AC-37(11):1666-1682, 1992.

[12] G. KREISELLMEIER AND B. D. O.

ANDERSON,

Robust model reference

adaptive control,

IEEE Trans. Automat. Contr., AC-31(2): 127-132,

February,1986.

[13]

M.

KRSTIC,

I. KANELLAKOPOULIS, AND P.V.KOKOTOVIC, Nonlinearand

AdaptiveControlDesign, John

Wiley, New

York,

1995.

[14] P.R. KUMAR AND P.P. VARAIYA,

StochasticSystems:Estimation, Identi

ficationandAdaptive

Control, PlenumPress (NewYork),1986.

[15] R. V.

MONOPOLI,

Model reference adaptive control with an augmented

error signal, IEEE

Trans.

Automat.Contr.,

AC-19:474-482, 1974.

[16] A. S. MORSE, Global stability of parameter-adaptivecontrol systems,

IEEE

Trans.

Automat.

Contr., AC-25(3):433-439, 1980.

[17] A.S. MORSE, Overcomingthe obstacleof high relativedegree,

Journal

of theSocietyfor InstrumentandControlEngineers,

34:629--636,1995.

[18] A.S. MORSE, D. Q. MAYNE,

AND

G.C.

GOODWIN,

ApplicationsofHys

teresis Switchingin ParameterAdaptiveControl, IEEE

Trans.

Automat.

Contr., AC-37(11):1343-1354, 1992.

[19]

S.M.

NAIK,

P.R.

KUMAR,

AND

B.E.

YDSTIE,

Robust Continuous-Time

AdaptiveControl by ParameterProjection,

IEEE

Trans.

Automat.

Contr.,

AC-37(2):182-197,1992.

[20] K. S.

NARENDRA, Adaptiveand LearningSystems-Theory andApplica

tions,PlenumPress (NewYork),

1986.

[21]

K. S. NARENDRA, Y.-H. LIN, AND L. S. VALAVANI, Stable adaptivecon

troller design, Part II: Proof of stability , IEEE

Trans.

Automat.

Contr.,

AC-25(3):440-448, 1980.

[22] S.

SASTRY AND

M.

BODSON,AdaptiveControl: Stability, Convergence and

Robustness,

PrenticeHall (EnglewoodCliff,NJ), 1989.

D.Q.M. & L.L.

86



II. PaOBUDlI STATBMENT

In tbiI paper we shall be coacemed with theadaptive control

of

liDear

time-iavariant

fiDite-dimeDlioDal

systemI

haviq

the lollowina state

apace repreaentation:

X(I+

1)-Ax(I)+B,,(I):

x(O)-xo (2.1)

1(1)-Cx(t) (2.2)

wha'eX(I), u(t),

y(l)

are the 11X 1 state vector, rX I input vector,

and

m x 1 output vector, respectively.

A staDdard result is that the

system

(2.1),

(2.2)

can

also

be represented

il l matrix IraetioD, or ARMA, form u

m.eaD-Iquue output

is

bouaded

wheDevcr

the

samplemean-square of

the

noiIe

is

bouaded.

HOWC\W,

the

.-.u

questioD

of

stability

remaiDI

uaauwerecllor atoebutic adaptive aJaorithmL

The study of discrete-time

determiIlistic aJaorithms is of

indepeDdent

intereat

but also

provides

iDsiaht

mto

stabilityqucatioDl

in the stoehutic

cu e (IS].

Recent work by 100000U

a d

Monopoli (13)

bu

been

concerned with the

exteDlioD of the

results in

(2) to

the cliIcrete-time

cue. AI

for the continuous

cue, the aupICIlted

error method is

used.

In tbiI

paperwepreIeIlt

11ft' resu1tI

nIatecI to clilCJ'ete-time

iatic

adaptiw

coatroL

Our approKh diffen from previous

work

in

several ~

respects

altbouP

ca1aiD upectI

of our

approach

are

iaspirecl by

the workof Feuer aDd

MOlle 5].

The

uaIysia

preIeIltecl here does

Dot

rely

upon the

use

of

aupDeDted

erron or awdlWy iD.putI. Moreover, the alpithms havea

Vf IIY simple

structure

and ue applicable to lDultipltHDput

multiple-output

systems

with rath. pnera1

UlUlDptioDI.

11le paper praenta

a paeraI

metbod

of

auIyIiI for dilcrete-time

detenD.iDiltic adaptive control alpithmL

The

JDethod is iDUitrated

by

.tablilbiDa

slobal

converpDCe

lo r

thJeesimple alpithmL

Fo r clarity

of

prIMIltatioD,

we shall

first

treat a

simple

IiDIIo-iDput siqIe-output

alpitbm

in

detail.The reaults

wiD

then

be

exteaded

to 0

siqIo-in

put aiqIe-output aJaoritlulll

iDdudiDa

thole hued on recunive least

squarea. Piully, die exteDIioD to DlaltipJe..iDput

multiple-output

systems

will

be prIMIlteeL

SiDce the results in

this

paper were

prIMIltecl

a

Dumber of

other

authon [16]-[18] have presented related resultl lor diacrete-time de

termiDiltic adaptive control

alpitbms..

Short

Papers

Dllaet&-11IIIe

_

Adapdve

Coatrol

GRAHAM

c.

GOODWIN,

. . . . . . .

JIIIB,

PETER. J.

RAMADGI , AND

PETER.

E.

CAINES. MJ I88I. IBI B

.4....

n i l Ur •••

c-..

..

.............. ' 6 I I c l

........

•••• rde

-..,--.It II

tIIIt wII-..

...... . ....

,

..

............................

I. IN'raoOUCl10N

A _

problem

il l

control theory hasbeen the questioD

01

the . . . . . . oIlimp1e.llot.lIy

COD t

adaptivecontrol

alpithmL

By t1IiIwe III8Ul aJaoritlulll which,

for

all

iDitial

I)'IteID aad aJaoritbm

C&1IIe

the outputl

of

a liveD IiDear

system

to _

track

a

daind

output IICIueDCet

aDd

achieve

this with

a bounded-iDput

IeqUeDCL

11lere is

a

CODIidcrable

amoUDt of literature

OD

continuous-time

determiDiltic adaptivecontrol alaoritbma[IJ.

However, it

is

0DIy

recendy

that aJobal stability aacl conv-aeace of these aJaoritlulll bas

been

studied UDcler a-aeraI

uaumptioDl.

Much

interest wa

paerated by the

iDDovative

CODfipration

pI'OpC)Ied. by

Moaopoli (2) whereby the feed

back

piDI were

directly

.timated and aD

aupleDted

error sipal aDd

awdlWy

iilput aipaII

were

introduced

to

avoid the use of

pure

dif·

fereDtiaton

il l

the aJaoritbm. UDfortunate1y,

as

pointed out

in [3J

the

8I'JUIIleD.tI

livea

il l

(2)

CODCeI DiDa

stability

are

incomplete. New proofs

for related

aJaorithma

have recently appeared

(4], (S]. In

(4]Narendra

and.

VaiavaDi

treat

the

cu e

where

the difference

in

orders between the

DUIUI'&tor

aad deDomiDator 01 the

system

traDller function (relative

etearee)

is laa thaD 01' equal

to

two. In (5], Feuer and Morse

propose

a

solution for

paeralliDear

systems without coDltrainta

OD

the relative

dep'ee. The alpithma il l [5] use the

auplented

error concept and

auxiliuy iDputi

U in [2J. The

Feuer

and

Morse result seems

to be

the

moat paeral

to

date for siDale-input siqle-output continuous-time

sya

tau. Howevw,

these

results are teclmicaJ1y involved and caDDot be

directly

applied to the dilcrete-time case.

11lere

hal

also been interelt in'dilcrete-time adaptive control for both

the determiDiatic ADd

Itocbutic

cue. This ana

baa particular

relevance

ill viewof the iDcreuiDa ut e of ctiaital tecJmolo&y in control applications

(6),(7).

ldUDI

[8],[9J

bU proposed a pneral techDique for

auaIyziDa

conver·

aence of diacrete-time Itocbutic adaptive alaoritbms.. However, in

this

auIyIiI a

q..aioa whichis ye t

to

be

reaoIved

ccmcems

the

boUDdednesa

of the . ... vuiabl-.

For

OM

particular

aJaorithm

[10], it

baa been

arpecI

iD [II] that the alpiduD poIIeII the property that the sample

(2.3)

with

appropriate initial conditioDS. In (2.3), A(q-I), BU(q-l)

( ;

1,··· ,m; j - l , · · · ,r ) denote scalar polynomials in the unit delay opera

tor q-l aDd the 'acton q- represent pure

time

delays.

Note

that it

is

not UIUIIled tbat the

system

(2.1),

(2.2)

is completely

controllable

or completely observable, DOl'

is

it

IIIUIDeCl

that (2.3) is

imclucible.

The systemwill be

required,

however,

to satisfy

the condi

tioDa of Lemma

3.2.

It is UIUIDed

that

the coefficientsin the

matrix.

A, B, C in (2.1),(2.2)

are UDbown and

that

the s ta te (t)

is

Dot

directly

m.surable.

A

feedback

coatrollaw

is

to be

dcaiped

to ltabilize

the system

and to

caue

the u t u ~

(y(I»,

to track

a liven reference

sequence

{y·(I)}.

Reprinted from

IEEE

Transactions on Automatic Control,

Vol. AC-25 , 1980, pp. 449-456.

487



Specifically, we requireye'l

and u(t)

to be bounded uniformly in I, and

LentmtJ 3.2: For

1MsyJtem

(2.3)with r - m, and subjectto

lim y ~ ( I ) - Y I · { t ) - O

i - I , - · ·

,me

'

....

00

UI.

KEY Tl cHNlCAL

LBMMAS

(2.4)

[

zdu-dIBII{Z)

det :

Z ~ I - ~ B I ( Z )

(3.6)

Our

aualysil of

discrete-timemultivariable

adaptive

control a1p'ithma

for

Izi <1

WMn

will be buecI OD the loIlowiDa

technical results.

UmmtI

3.1:

11

14- min

tlu

I <j< ,

i - I , · · ·,m,

(3.7)

ax 1 ) , t + 4 ) 1 - ~

O<t<T

1<1<111

if

'11M , . , . u in COfUItUttI

m,.

4

whicll

an of T witA0 <III ] <

00 ,

0

< 4< 00

.rudI

that

hoof:

The rcault

ilataDdard ad simply followsfrom the

fact that

(3.6)eaaunI that the systemhu • stable inverse. 0

IIIthe

remainder of the

paper thae

reaul.

willbe used to

prove

Blobal

CODV...... of a Dumber of

adaptive

control aJaorithma. Sections

IV-VII will be concemed with

adaptive

control of

siqle-input

output

ayatemI.

SectiODI

vm and

IX

will

extend

these results

to

the

mul1iple-iDput

m u l t i p ~ u t p u t

cue..

(3.1)

(3.2)

(3.4)

(3.3)

wIleN0<CI <

00 ,

0<C

2

< co,

i l follow8 tMt

lim 1(1)-0

'

....

00

O<b.(t)<X<ao

and

O<b

2

(1)<K < oo

for

aliI>

0 tIIIIl

2)

COIIIlltitIII

lIer(t)H <

C,

+

C

2

max I.r(

1')1

0<.,.<,

lim s(/)2

-0

-+00

b l I ) + ~ t ) c r / ) T c r )

w1Mre (b.(I», ~ I » ,

IIIttI

(aI(/)} t ire retIlXG1tIr tIIItl (CJ(t)} i.r II

1WIIJl-tJ«IfJr .....,.eei . .

Alb}«1

10

1)

f I I I i n I I ~ CtJIIdltlort

tIItIl {UCJ I)JI} i8

bouItdttd.

Proof: II (I(/)} ill

a

bounded

sequence,

thea by (3.3) (IIcr(1)1I) is

a

bouDdecIleqUeDCe. Thea by (3.2) and(3.1) it follows that

lim 1(/)-0.

1-+00

IV. SINGU-1NPtrr

SlNGu-otrrPur SYSTBMS

It is

well

known

that

for the siD&le-input single-output(SISO)case, the

system output of (2.1), (2.2) can

be

described by

(3.S)

ad

Hence,

(4.1)

whero (u(t)}.

(yet)}

denote

the input and output sequences, respeo

lively, mel A(q-I) . B(q-I) are polynomial functiODS of the UDit delay

operator q-l ,

..4(q-I)-I+G,9-

1

+

.. . +1I,.q-

B(q-I)-bo+b1q-I+

.. •

+b ,q- '; bo+O

d

repJaeJltl the

system

time delay. The initial conditions

of

(2.1) arc

replaced by

iDitial

values

of

y(t),

0> t > -11,

and u(

t).

-

d >t»

-

d

-

m

The

loIlowiq 'UI1IIDptioDl will

be made about the

system.

A ru1It tiM Set 4:

a) d is Imown.

b) AD

upper

bound

for

and m

is

known.

c)

B(z)

hal all zeros strictly outside the closed unit disk. (This

is

neceaary

to easure

that the

control objective can

be achieved

with a

bounded-input

sequence.)

We

note thaltby

successive substitution, (4.1)

can

be rewritten u

> x

1/2

+X

I

/

2

U

cr(

t,.)

fI

:>

18(1,,)1

gl/2+ KI / ,

C. +

C

2

1r(t,,)

1] uaiDa (3.3)

and

(3.5).

lim

lor(

',.)1- 00

1,.-+00

Now IIIUJIle

that

(.t(t)}

is unbounded.

It follows

that

there

exists a

lubiequeDce (I,,)

such that

I.r< t)1

<

'8(,-)1 for

I <I .

Now aloDa

the

aublequenee

{t,,}

I

.r(t.) I> 18(1.)1 uaiDa

(3.2)

[

b. (

..>

+

1.)_(

t,.)T

G( . ] 1/'1 [ K

+

KII_(

',.)112]

1/2

lJ(t,,)1

where (y·(t)} is a reference sequence.

It

is assumed that (y·(t)} is

kIlowD

II

priori

and

that

lim

I .r(tJ I> _1_ >0

,--+ao

[ b . I , , + ~ I , , a t , , : r c r t l l

]1/2

K'/2('2

but this contradicta (3.1) IIld heDce the assumption that

(8(/)} is

un

bounded it falaead

the

remit

follows.

0

ID

order

to 1110 this

lemmaill

proviDa pobal CODveqence of

adaptive

control

alpithma

it

will be necessary

to

verify

(3.1)

(with .r(t)

interpre.

ted

u

the trackiDa error) aDd

to check

that IlllUlDPtioDa (3.2)

and (3.3)

are satiafied.

The nat lemma

will

be

used to

verify

that

the linear boundedDCSI

conditioD

(33)

is

.ti Ified

by an important e. . . . of linear timo-invariant

ayatemI.

This c1uacomspoDda to

tho.

tiDear timo-invariaDt systemI for

which the control objective

(2.4)

CU l

be

achieved with a bounded-input

sequence aDd lor which the trackiDa error

ca D

be reduced

to

zero i f

the

systaD

parametenare known.

y(t+ d)-a(q-

J)Y(/)

+ /l(q-I)u(t)

where

As previously

stated, the control objective is to achieve

lim [Y(I)-

y·(t)J-O

' ....00

(4.2)

(4.3)

(4.4)

488



V.. SISO

PROJECTION

ALGoRITHM I

Let .

be the vector of systemparameters (dimensionp - n

+

m

+d).

for all values of tp(1

- d)

provided

0 <0(/) <2..

This

is

satisfied by

definition

(5.8). Then,

since

111(/)11

2

is

a bounded nonincreuing function

it converges.Setting

(S.I)

(I). -

,,(1- d)T;(/- l)

(S.13)

lben

(4..2) can be

written

y(t+d)_cp(t)T,O

where

(5.2)

[and DOtin.that

tI(I)[

-2+11(1)

, , (I-d)T,(I-d)

]

[1+f)(t-d)7 cp(t-d»)

Now

doIiDe

the output trackiDI error as

,<1)7'

-(y(t),· ·

. ..,(/-11+1), (/),··

·

,11(/-

m - d+ 1».

(5.3)

is boUDded away from zero, with a(/) defmed u in (S.8») we conclude,

from

(5.12),

that

+ d)

-

y(l+ d) - ,·(1 + d)

•

cp(

I)

T

0 -y.( 1+d).

By chOOliDa

(II( t)}

to satisfy

cp(/)T O·y·(/+d)

(5.4)

(5.5)

and hence

(5.14)

it

is evideat that the tractiDI error is

identically

zero.However,

since

0 Now using (5.13)and (5.11)

it

follows

that

is UDbaowD, we replace

(5.5)

by the

foUowinl adaptive aJaoritbm:

(5.9)

(5.10)

(5.15)

e(l)

[ 1+cp(t - d)

T

«p(

1-

d) ]

111

4-1 - d)T

-

0(/- ;) y\/

i- I [I

+4fJ(t-d)T.,(t-d)]1/2

, , ( I -d- i )

e(I)- _cp(l-d)Ti(,-d).

(I)

0<I 0 / - i)tp(t- d)T fP(t -d- ;)

[I +

'<,- tl)TCP(t_d)]1/2

[1+. t -

d-i T. , t -d- i)]1/2

f(1-1)

I

Hence,

[1

+cp(t-d-i)T <t-

d -

;)]1

12

• (t - i) . (5.16)

[I

+cp(t-d-

i)Trp(t-d - ;)]1/2

Nowby the Cauchy-Schwarz inequality and the fact

that la(t)1

<2

. f (1-d{ < I -d- i ) f(l-i).

[I

+.ct-d-i)TqJ(t-d-i)]

Then uaina (5.4)and S.7) wehave that

d - I

rz

«(I) . -4p( t-d) I ( / -d ) - 11(1-

i)

I- I

· [1+tp(/-d-

i . c t -d- ;»)1/2

I

2 ( 1 -

i) I

< [I+cp(I-d- j)Tcp(I_d_ i»)1/2 •

Then using (5.14) it follows that

7',-

lim tit)

1(/)

-0.

I--.CO

[1+9(t)T4p(t)]l/2

Proof: UIiDa the

dermitiOD

of i(t),

S.6)

may be rewritten as

i(t)

-

i ( / - l )

-

a(t)4p{t

- d)[l

+q>(t- d)T.(1

-

d)J-l

·,,(t-d)Ti(t-l). (S.lt)

i(t) - if1-

I)

+a(t).(1- d) [

1

+ .p(t - d) Tq> 1- d)]-1

T.

A

]

· [y(t)- cp(t-d) 1(/-1)

S.6)

tp(1)Ti(t)-Y·(I+d)

S.7)

where

i(t) is a

}I-vector

of reaIs depeadiDaon an

initial

vector i(O) and

ony(t'). 0<.,<1. u(1'), O<.,<t-dvia (5.6),

and

wherethe p in CODItaDt

a(/) iJ

computed u follows:

a(/)-1 if [(II +

I)th

component of right-hand side of

(5.6) (5.8)

evaluated

uaina

a(t)-I ]+O;

- y otherwise where y is a constant in the interval

(e:.2-e:). y. .1 and 0<<<< I.

Thischoice of pin constant prevents thecomputed coefficientof u(I)

in S.7) beiDa zwo.We

also

remark

that

the purpose 01 the

coeffICient

1

in the

term [I

+f(t -d)Tf ( / -d)]-1 of

S.6)

is

solely

to avoid diviaioD

by

zero.

ADy

positivec:oDltaDt could be used in place of the 1.

Apart

from

thellbove modificatioD,

the

al.orithm

(5.6)

is

an

orthogo

lUll projection

of1(/-1) onto

thehypersurfacey(/)-tp(t-d)T,.O.

In theauJyaiaof this alJorithm, we

wiD rust

show

that

the Euclidean

Dorm

of thevector i(,)- 1< - '0 is a

noniDcreainl

fUDCtion alODI the

trajectories 01 the a1lOrithm. This leada to a characterization of the

limitin. behaviorof the alaorithm which will allow us to use Lemma3.1

to establishpoba1 coDveraence.

lAmmJI

5.1:

Along

1'-

801ution.r

of

(5.6), (5.7),

Hence,

;(1)11

2

- Ui(t-l)tf2.tJ(t)[-2+11(1) ,,(I-d)TCP(I-d) ]

[1+.,( t - d)T

t - d) ]

.

i (1-I)Tf( I-d)f I -d)Ti( , - I )

<0

(5.12)

[I

+.p(/-d)T,,(t-d)]

, , ( t -d- ; )

[1

+,,(

t

- d - i)

T

q>( t - d_ ;)]1/2

. (1 - i) 1-

0

for

;-1,2, .. ' td-1.

(S.l7)

[I +

, ,(1-

d -

;)Ttp(t_

d - i ) t

/2

489



but

TberefOl'et l I I iq (5.3)

lIt'<t-d)1I

<p{

m,+[max(l,mJ)

max

IY(1 )I}

1<..

<1

hotI: Lemma

5.1

CDIlII'eI that condition

(3.1)

of Lemma

3.1

is

aatiafiecl,

with 1(/) - e(/). the traekiq error,

and

0(1)-

cp(

1-

d)

thevector

defiD.ed

by

(5.3).

Also

6.(/)-1,

and 6

2

( / ) - I .

It

foBows that the UDiform

bouDdedaell

CODdition (3.2)

is

satisfied.

AasumptioD 4c) a d

Lemma3.2 ensure that

Ju(k-tI)f<m,+m.

max

IY(1')1 foralll

<k<t.

1

<or<t

(6.3)

However,

since

'0 is UDknown,

the

control law will be reeursively

estimated. The foUowina adaptive

aJaoritbm

will be considered:

i t - i t -d - ~ . , t - d [ l

+cp(t-

d)T <t-d»)-1 e(t) (6.4)

Po

( t)

_.,(/)Ti(/)

(6.S)

whereA, is a

flXecJ CODStaq,t

and

i(t) is

a p-vectorof reals

depending

on

d

initial

values 1(0),•• • , I (d- I ) and on y(.,), 0<., cr , u(T).

0<1 <

t

-

d

- 1

via

(6.4). Not e that (6.4) is

aetuaI1y d

separate recunioDl

interlaced. (It bas recently been pointed out (18] that it is also possible

to

analyzea _ p e recursion without

iaterlaciDa lII iq

a

different

technique

but the same aeneral priacipals.)

The aualysia of projection

a1aoritbm

n bas mudl in common with the

analysis

for

projection

alptbm

I.

We

wiD

therefore

merely state the

analop

of Lemma S.l

and

Theorem S.I for the algorithm (6.4),(6.S).

ummtJ 6.1:

DejiM

where

cp(t)T

- (

-y(t)_·· -y t -n+ 1), - u / - l · · ·

-u t -m-d+ l),y·(t+d»

,T _ (ft -

P

R' I )

o - . ••• ,«,,-It

I · · · ~ d - I

Po .

It

is evident that the

traekiD

error can be made

identicaUy zero

by

ehooaina

(u(t)l such that

(5.19)

im

[Y(/)-

y·(t)]-O.

t-.ao

Hence,

usiq (5.16),

(S.I7), and (5.14)

lim

e(l)

-0.

(S.18)

1-+00

[I

+e,(/-d)Tf {/-d)]1/2

Thiaeatabliahea

(S.IO).

0

Note that

we

do

Dot

prove, or claim, that i</)

converps

to 0

However,

the

weaker condition (S.lO) will

be

sufficient

to establilb

CODV eoce of the trackiDa

error

to

zero

and

boundedneu of

thesystem

iDputiand output&. These are the prime

properties

of concern in adap

tive control

77Ieorwm $.1:

Subj«1

to

AUIIIPf'liOlU 4a)-e);

if

tIIII

iIlgoritlun

(5.6),

(5.1) II f/IIIIIW to lite1Y.J/eIn (2.1).(2.2)( , . m-l) ,

tMn (y(/» tIItd

(u(t»

tIIW boIwI64 tw l

; / -

i( /) - 'o.

(6.6)

_

..

0

TIJsJ

IIlI(t+d)U

1-1I1 ( / ) l f

z

<

0 aJOIIg witla tlw

lOhdioru

of (6.4) and (6.5)

tmd

Hence,

l I , < t ~ d 1 I <JJ{m,+ [max(ltmJl

max

[Ie(r)l+mlJ}

1<.,<1

-C +C

2

max 1e(1')I; 0<C.<oo,O<C

2<oo

1<.,<1

and

it follows that the

linear

bounc1edness condition

(3.3)

is also satis

fied.

The

reault

DOW foDows by Lemma3.1 and by n o q

that boundedness

of {II

,(/)II)

eIIIURI

boundedDessof

{IY<

'>I}

and

{IN(t)I}·

0

IJo

0<

-: -

<2.

Ilo

(6.7)

o

We

Dote

that the condition 0<

flo/A,

<2

hu

been previously conjee.

tured (9), (10)

in

reprd to

stoehutic self-tunina

rep1aton uainlleast

squares. The condition

ca D

always be satisfied if the

sign

of IJo and an

upper bound for

the magnitude of Po

are

known.

Lemma 6.1

is

used to prove Theorem. 6.1 in the same lIl81U1er that

Lemma S.I is used to establish Theorem

S.1.

We obtain

the

foUowina

theorem in

this

way.

'1'heomn 6.1: Subject to AUIIIPf'tioIu 4a)-c) and for 0<Pol

<

2;

if

1M

tllgoritlun (6.4). (6.5) i.r app/i«J to

lite

qslma (1.1), (2.2), (yet)}

and

(u(t)} an bounded and

(7.1)

(6.6)

im

[y(t) - y·(t)J-O.

' ......00

VII. ADAPTIVE CoNn.OL

USING

R.Ect1RsJvE LBAST SQUARES

The

wide-spreaduse of recursive least squares

in

parameter estimation

indicatesthat it

may find

application

in

theadaptivecontrolcontext We

treat the unit

delay

cue 01algorithm I with the projection (5.6) replaced

by recursive least squares.

The adaptive control algorithm then becomes

; t - i t - l + a(I)p(t-2).(t-l)

[1+a(t)cp(/-l)Tp(t-2)cp(t-l)]

[ y( t) - cp t - 1)T

i(

t - 1) ]

e(/+ 4)-y(t+d) -

Y·(I+d)

-

Po(

lI(t)+crQy(t)·

••

~ _ I y t - I I

+

1)+

I

11

( t - l)

... +1I.:.+4-1 (t-m-d+

1)-

~ y . t + d »

-1Jo(

U(/)-

.,(1)7'0)

(6.2)

Let

VI. SISO PaOJBC1 lON

ALoOIU1llM

II

In this seetioo we present an algorithm differing from that of Section

V

in that

the cootrollaw

is

estimated dirccdy.This approach

is adopted

in [5), and essentially involves the factorization of fJo from (4.2). A

related

procedure

is used in the self-tuDing replator (10) where it

is

assumed

that the

valueof

IJo is mown.

AD advaDtqe

of

the aJaoritbm is that the precautions required in

Section

V to avoiddivilioo

by

zero in the calculation

of

the input are

DO

lonler neceaary.

However,

a disadvantap

is

that additionalinformation

is

required; specificaDy, we need to know the sign of fJo

and

an upper

bouDd lo r

ita

mapitude.

PactoriDl flo from (4.2)

yields

y(t+d)-JIo(crOy(t)+

.. .

+c(_ly(t-n+

1)+11(/)

+

lJ;u(t-I)··· +

fJ:,,+d_I (t-m-d+ 1». (6.1)

490



(7.13)

P t -[ l - P(t-I),,(t)f(I)Ta(t+ 1) ]P(t-l) (7.2)

1+CP(t)TP(t-l)cp(t)a(t+1)

.(1)Ti(t>.y·(t+ I) (7.3)

where

p(t) is . Xp matrix and the recanion (1.2) is

8II1UIled

to be

iDitialized with p( -1) equal to any positivedefiDite matrix.

The ICI1ar G(t) in (7.1), (1.2) playa the samerole u in SectionV and

is

required 0D1y

to

avoid the noapDeric pOllibility of divisioD by zero in

(7.3)wh.

evaluatiq II{t).

Hence, G(I)-1 will almoIt always wort IDd

for tl(1)-1 we

observe

that

(1.1) aDd

(1.2) are the .tuldard

recursive

leut

IqlIU'eI

aJaorithm.

The sequence

{tI{t)}

may

be

choIeD

u

in

5.8).

u.. 1.1: AItmgwith 1M

8OIutItH&r of

(1.1), (1.2), (1.3)

.

/tIItctiolt

Y(1)-i(t)TP(t-l)-li(t) U

II bount/ed,

IIOIIMIt'tiN, PJOIIinc1W filfr jwIc

tioII

tIIIIl

Now

1,(t)1

2

le(t)1

2

[I+a(t)cp(t-l)T

p(t-2)cp(t-I)]

;>

[I+2U.(I-l)1I2<A.aIP(t-2)])]·

(7.12)

Hence,from (1.11) and

(7.12)

lim 1 <1)1

2

-0 .

''''00

[1

+2(Up(t-2»))U«p(t-l)U

2

]

TbiI

will

be

recopized

as

beiDa

condition

(3.1)

with .f(1)-

ce(t),

b.(I)

1, and

bi )

- 2(A..,J.p(t

- 2)D.

To establish the UDiform boundednesa condition (3.2) we proceed as

foUows. From (7.2) and the matrix inveniOD lemJDa,

r-

lim

.(1-1) ' ( I - I )

-0

'-.00

[1 +a(t)cp(I-l)7 P(t-2)CP(I-I)]1/2

i(t)-i(/)-,o-

Proof: Prom (7.1),

S.2),

• ...

a(I)P(I-2)tp(I- l )cp(t- l )T;(I- l )

1(1)-1(1-1)- .

[I +tI(t)CP(I-I)Tp(,-2).(t-I)]

Then uain

(1.2),

(7.4)

7.S)

pet) -1- P(I- l ) - I +a(t)cp(t)9(/)T.

Hence,

XTp(t)-l

x

>xTp(t - l )x

>A.ua[p(/- l)- l l

llx

Il

2

for each xEA'. (7.14)

Now

choose x as the eigenvector corresponding to the minimum

eigenvalue of

[P( t) - I

Then

from (7.14)

; ( I )_P(I- I )P(I-2)- I ; ( / - I ) .

Thus,

P t-l -I; I -P t-2 -Ii ,- l .

Now defllliD&

V(I) U i(,

-1)TP(t

-1);(1

- I ) wehave

A.m[P(I)-I] >A.m(P(t-l)

-I].

So AaJp(t)

-I]

is a nondecreasing function bounded below by

A-JP(- I ) - I ] -X -

I

>0.

(7.6) Hence from (7.13),

o<bz(l)

<2K. This establishes condition (3.2).

The

proofDOW

proceeds

U forTheorem

5.1.

0

u(t)

(8.1)

q

-tC-.B.....(q-l)

VIII. MULTIPU-1NPur

MULTD'LB-<>urPuT SYSTEMS

For the case m- r» 1, the system (2.1), (2.2) can be represented in the

form

where Ak(q - I)

and

BIc/ q - I) 1<k <m, I <1<;m are scalar polynomials

in the unit delay operator q - l with nonzero constant coefficient

Using

the m

identities l -A;(q-I )} j (q- l)+q-4Gt<q-l)where

(7.8)

• T

r-

lim 1(1-1)

.(1-1),(1-1) (1-1)

-0 .

,-.00 [I

+ tI(t)4p(t-l)T

p(t-2)cp(t-l)]

lim e(/) -0

, .....ao [1+a(t)cp(t-l)T

p

( I - 2)

cp

t - I»)' /2

Hence,

Y(t)-

Y(t-l)_;(t)T

p( l - l ) - I ; ( t )_ ; ( t - I )T

p

(t - 2) - Ii ( t - I ).

UI iq (1.6)

Y(t)- Y t - I -

[i I - i t - l ]T

P(t-2)-1i( t - l )

• •

_

-a(t)

i(t-I('PC

t - l7<t-I)Ti(t-l)

(7.1)

[1

+41(1)'<'-1)

p(t-2).,( I-

I)l

wherewe have

used (1.5). It is clear from (/.7) that V(I) is a ~

DODDeptive, DODiDcreuina

function and hence converp8.

Thus. from fl.1), aDd since

a(t) is

bounded away froJDzero,

where

and

(7.9)

4-

min

(dil}'

i-I,···,nI,

1<)< .

o (8.1) can be written

1'II«Jmn

1.1:

Subj«t to Amlnpliolu 4tI)-c) if I. aI,orilltm

(7.1), (1.2),

1.3) i8

. Ii. to 1M

IYstem (2.2) (r- m-l) ,

IMn {Yet)},

{fI(I)}

are

botI1ItJ«J

and

Proof.

From Lemma7.1

lim e(l) -0 .

(7.11)

''''00 [1 +a(t)cp(t-l)7 P(t-2)9(t-l»)1/1

lim [.,,(1)-)'-(1»)-0.

I-.ao

(7.10)

[

Y l t ~ d l

] _

[ a l { ~ - I ]y(t)

,.(1+

tJ...)

0 a .(q-l)

[

~ 1 I q - l ...

+ :

I3lftl(q-l)

491



f<a(t)<2-f

where O<e<1 and G(/)-l is no t an eipavalue of - R

- 1 ( ~ - l ) Y ( / )

with

where ~ ( t ) is -P,

( -

+

m(m,

vectorof realsdepending upon an

initial

vector

',(0) andy,(.,.), 0< <I,

u(T),

0<,. <1- d,via (9.4).

Oearly, it is critical

to

eusure that a solution to 9.S) exists for all t.

This

is guaranteed if the matrix of coefficients of u(t) in

(9.5)

is

noDliDplar this is

eDlured

by the procedure.

1) At

1-

0 the

arbitrary

initial value 1(0)

of

the

parameter atimate is

choeea. 10 that AuumptioD Set a is satiIfiod.

HO lce,

the aDaloaOUI

equatioDa to

(9.3)

evaluatedat

4 f ( 0 ) , y ( ~ ) ,

J< I < ..1(0)

He

solvablefor

11(0).

2) Fo r

I:>

1 the proceduJe of Lemma 9.1 below guarantees the 101va

bility01 thealgorithm equatiou for II(I).

Um1IIII

9.1:

I

ortIttr

tItat 1MIIttII1U t1/

cM/fidm/8

of

u(1)

in (9.5) i.r

1tOfLfbIIultv

for

Gill:> I U

u

III/fIdett

for

a(t) I (9.4) 10

be

cIIomt 4r

jollowl:

(9.6)

(9.7)

(9.8)

(9.9)

T ; - S, ,(t-I)

and

Y(/)

ro.,··· '011I]'

'. in (9.6)

is

the vectorof coefficientsof

II{t) in

,{ t

-

1), that is,

when

, in (9.6) is the vector of cbaDps in the coefficient of 11(/), that is,

where

and

, ( q - l ) - I i ( q - I ) B U ( q - l ) q 4 4 - ~ . 8.3)

It

can be seeD that

(8.2)

consists of a set of multiple-input siDIle-out

pu t

(MISO) systems haviDa a common input vector.

The

foUowiq

asaumptioDl willbemade about the system.

A .tltHt Set B:

a) d

1

, - · · ,tl.

are known.

b)

An

upper

bound

for the order

of

each

polynomial

in

(8.2)

is

knoWD.

c) The system(8.1) satisfies condition

det[

,lIIu-.t'Bu(z) .. • zd••-tllB. ,(Z)] ....o

rr: for Izi <1.

z4a-4l(.B..,(z)

•. •

4. ....B....(z)

Condition Ie) daervea commenL

Firat, for ally output colDpOnentYI' 1<i <m, there exists at least one

polynomial

9 - ~ B u < q - l ) .

I<j<m, for which the power of , -

associated with ita (DODZefO)

leadinl

coefficient is 4-

For

each such

po1yDomia1 theuaociated input appean in

y,(

t with the least

possible delay4.Evaluated at z-O condition Ie )

requires

the matrix of

these1eadinl coeIfic:ienti to be

1lODIiDp.1ar.

S e c o D ~ the aenericity of condition Be)(for the model (8.1») depends

upon

the

ini tial parameter ization of the system from

which

(8.1)

is

computed. TbiI cIepeadence is currently under investigation.

The control objective, u before,

is

to achieve

lim

[y,(/)-

yt(t»)-O

i - l , · · · ,m

t .....ao

IX. MIMOADAPrIVB CoNTllOL

where (/ ) is a reference sequence. It

is

assumed that each (yt( t )}

is 0,-

Sir C P I ( t - ~ ) ( l +9,(/-

~ ) T c p I ( t -

~ » - I

known

a

priori and that

ly,·(/)f <ml

< co for

aU

I, 1,- _.

· ( , , ( / )-

, ; ( t - ~ ) T I ( 1 -1»)]. (9.10)

(9.11)

1

<i

<em.im 1.1;(

t)

- y/.(1)1-0;

' ....00

Proof: Usins

Lemma (5.1) for each

i,

we have

. e.(t)

lim

f

-0

1/2

•

' ....00

[1 +

41>;(1

-

cp;(1 ]

Then:

i) R(O)is nODlinpiar by the initial choice of ; ~ O ) , ; - I,

. . .

t

m.

ii)

Assume R( I - l )

is

nODSinplar. Then from (9.11), usinJ a(t)+Ot

detR(t)-[detR(t-l)][det(l+a(t)R(t-I)-1

V(t»]

-ldetR(/-l»)(Q(/» [ ~ / » ) I + R / - I - I V(t)]

- 0 if and

ODly

if at/)

is

an eigenvalue of

-R( t - l ) - I

yet).

But the defini tion of a(/) euures a(t)-I is not an ei&envalue

of

- R(t-l)-IV(t),

hence

R(/)

is nonsingular

.. However, by i) R O)

is

nODSingular

and it foDows by induction R(t),

1

;>0 is noDSiDplar. 0

We note that the above choice of a(t) hu been included for technical

completeness

and that

a(

I) -

I

will

almost

always

work

since it is

a

Dongeneric occurrence for t to be an eigenvalue of -

R(/-l)-IY(/).

Also since - R(

1 - 1)- I

Y( t) bas only a fmite Dumber

of eigenvalues

it

is

always possible to find an a( t) to satisfy the lemma by computation of

the eigenvalues of R(1- 1)- I Y(I).

I Morem 9.1.·

Subject to Ammption.r &I)-c) if tlte algorithm

(9.4),(9.5)

if

applied10 tlw system

(2.1), (2.2)

with

r=m,

then

(y(/)}

and (II(I)} an

bou1ukd

and

(9.1)

(9.2)

(9.3)

<i<m.

· 1 + ~ ) - Y I t + ~ ) - , t t + 4 )

_ ~ ( t ) T ~ _ Y : ( / + ~ ) .

i , ( t ) - ~ ( t - l ) +a(t). , (t-d,)[ 1+ C P i ( t ~ t 4 ) T , , ( t - d,)]-I

T )

· ( Y I ( I ) - C P i ( t - ~ )

; ( t - I ) (9.4)

c p , ( t ) T ~ ( t ) _ , , { / + 1

«t <m (9.5)

492

Defme

where

It

is

evident

that

the

trackiD

error

may

be

made

identically

zero

if it

is

possible to choose

the

vector

u(/)

to satisfy

Let

'&

be the vector of parameters in (1,(9- . ) an d

fJll(q-I).

• • fJ,.(q-I).

Then (8.2)

may be written il l the

form

,(1+ cp,(t)T, . 1

<;

<m

Proof: The proof

will

be by induction and we first observe that

This section

wiD

be concerned with the multivariable versionsof

the

from (9.4)

and

(9.6)-(9.10)

adaptive control algorithms introduced in Sections V and VI. The

multivariable

version

of the allOritbmof SectionVII

also foDows analo- R( I) -

R(

1- I) +

a(

I) V( I).

gousty.

A.

MIMO

Proj«tioft

Algorithm I

Obviously (9.3) is a set of simultaneous equations in ..(t). Now the

matrix multiplying 11(/) is

nonsinplar

since in (8.3) det(diaIJi(z»-t at

z-O and Assumption

Ie ) holds. ~ e n c e

a

unique

solution

(1)

of (9.3)

exists at the true parameter value IJ.

Consider the

foUowilll

adaptive algorithm:



lim II,(/

..

)U-

co

I..

The proof

DOW

followa

that of Theorem. 5.1,except in the case that the

vector

)'(1) is unboundecL In

this case there

exists

a subsequence

(I,,)

such

that

and

IY/(

I + < IYJ('-)('- + forsome

I <al(

t..)<m

,aacl for all; I <I

<m and

tor

alii

<t,..

It

thea follOWl

by Lemma3.2 that

there

exist

coDltanti O<C

1

<co

IDd

O<C

2

<

oo

lUCIa

that

11.,('..)

< C I + C 2 I Y J , - t , . + ~ , . » l t

I <I<m.

SiDce

m is

fiDifet

there ail . . a further subsequence

{t,,}

of

the

sub

MqueDC8

{III} .uch

that

11., ,.,)11

<C.+C

2

Iy,(t

.

+4)1

forat

least

one

i, 1

<i <m

&lid (1,(,.,+4)} is ubouDded. The remaiDder of the proof then followl

dlat

01

Tbeonm

5.1wherewenote

that

where

d-max(d

1t

• • .Jt(J. i(t)T

is

aD ntXn

matrix of reala

depeactiq

OD II

iDitial

matrices I(f)T, 1

<; <

d aDd put data from

the

ayatem. P

isa

matrixof

COIIItaDII specified

aprlorl. AI in Section

VI.

<9.14)

repraeats

d interlaced recunioDa.

0DcI the recuraioDI (9.14), (9.1S) are iDitialized in a JDIImet thatIeadI

to • aique

IOIutioD f«

u(0) it illUlficieDt

to

e.uure that aD IUCCaliw

recunioJIIlead

to equatioallOlvable for

u(t),

I> I. In ..... to Lemma

9.1

we

have

foDowig. ..

U1nmtI 9.2: /hfttte I(t+tl)T _l(t+d)T- 'OT tIIId I«

K -

pro-

q

gT+K - KTg i.rporitiw

dI/iItit-,

t , tIlotw 1M tTtIjectorla

of (9.14),

(9. /5):

a)

trace[

i(I+d)Ti(t+d)]

-trace[

i(t)Ti(t)]

<0.

b)

lim e , t + ~

-0 , I<i

<me

t....

[I+,<I)Tcp(I)]I/2

Proof:

a) We caD

rewrite

(9.13)

UIiD

(9.1S) u

o

e,(I)I-IY,(

t) -

yt(

1)1·

Heace, from (9.14)

B. NINO Pro}«tioIJ Al,orltllm I I

From (8.2).

(8.3) the

fact

that

1'j(z)-1

for

z-o.

i - I , ·

. . , lit uad by

AuumptioD

Ie)

wecan

factor

out

the nonsiDgular

matrix

r

o

(-(IJ,(O)D

and

givinl

o

«t

<m.

X.

NON'LINBAIl

SYSTBMS

lim IY,(

t)

-

y;e(

t)I-O,

1-+00

or

trace(i(1+tI)Ti(t+ d)

-trace(

i(I)Ti(t»

_-trace[(KT+K-KTK f(t)T,(t)

)

(I+.<t)Tt(I)]

.(

i(t)T.,(t)[1+.,(t)T.,(t)r l.ct)Ti(t»

]

<

0 if gT+K

- KTK is positive dcfiDite.

b)

ill

the

proof

of Lemma (7.1)

it

foDows that

tim traee(x

r

+K-XTK

f(t)T,(t) )

1-+00

[I +.<t)I''<I)]

. ( i(1)TCP(t)[1 + cp(t)Ttp(t)]-l.,(t)Ti(t»_O.

SiDce

gT+

K - KTX is positive

defmite

then

tim

roll . ][1+9<t)r.,<t)J-

1

( t l ( l+

d) ·

.. e.(t+tJ...)]rol-o.

''''00 ·

-..(t+t.(.J

ThiaiJDpliea that (9.17) holds. 0

UIiDa

Lemma9.2and foDowiDa the proof of Theorem 9.1we havethe

folJowiDl.

1 II«nm 9.2:

Subject to

AulurptioJu

&I)-c), and K

T

+

K

- KTKpo.r;

ti w

definite, if

Q/pritltm (9.14). (9.15) ; appli«l to

1M.rystem

(2.1),

2.2) willir- m, ,hDJ

1_

fJ«10I .J y( I)

awl

II(

t)

tW bormd«IlIIId

Altho. the

aaalysia

in the

paper

baa been

carried

out for determinis

tic

Jinear

systems, it is clear that

it

could

be readily exteDded

to

certaiD

cIuIea of

nODlinear systems

of bowD form. The essential points are the

form 01 (5.2) or (6.2),

and the IiDear bound CODdiIiOD (3.3). The latter

point

would indicate that systems with cone bounded nODlinearitiea

(9.1S)

would satisfy

the

conditioD&.

493

Define

aDd

where

[

]_[JlI t;d

l

) ]_[JlW;d

l

) ]

e ,(t+

tI..)

y ,(t+tI..> y :(/+ t(..>

[

[

yf(t+dt) ]1

- r

o

u(t)+C(q-l)y(t)+D(q-l)u(t-l)-r;1 :

y':(t+d,..)

- r0<u( t) -

'OT.<

t»

(9.13)

where

'oT

ia

an

X,,'

matrix

whose

itb

row coataiDa the

parameten

from the ith fOWl of C(q-I). D(q-I), and E - r

o'.

f(/) is an

,,'X

1

vector

coataiDiDa

the appropriate delayed

veniou

of

y(t). u( t

and

y·(t):

4p 1)T

_ (

_ y(t)T, _

y t-

J)T,•. • , - fI(I_I)T,

-U(t-2)T, . . .

,yr(t+d.),···

,y,:(t+tJ..».

ADaIoaoUlly

to Section

VI we

introduce the followina adaptive control

alpithm,

[

el(t+d.) ]

i(t+d)T

_i( i)T-p : [l+ P(t)TfJ(t)]-I.(I)T

(9.14)

_.(1+

d .)

lIe

t) -

i(

If ,(t)



xr, CoNCLUSION

The paper

hu

analyzed a

pneral class

of

discre.time

adaptive

control alaorithma and

has

abown that, under suitableconditions,

they

will

be

alobaI1y convcqent- The

alaorithma

have a

very

simple structure

and are applicable to both

sinl1e-input

siD&10-0utput and multiple-iDput

multiplo-output systemswith arbitrary time delaysprovidedonly that a

,table coatrol law exists to achieve zero trackiD error.

The

results

resolve a loDa staDdiDg question in adaptive control

reprdina

the

existence

of simple. atobally

converpnt

adaptive algorithms.

(1) I. D.

LaDdau.

A

lurvey

of model refa'OllCC adaptive tec1miqua-Tbeory aacl

appticaltall.

A

........

,

vo l to.

pp. 353-319,

1974-

(1) R.. V. MoaopoIi. Model

ref..-

adaptive

COIltrol

with al l a teeI error

Ii

lEU

TMu. AI.. . . , . e-tr., vol. ACI', pp. 474-415,

OcL

191...

(3) A. P ,

8.

R.

BarmiIb,

aDd A.

S. Mone. NAD astable dyDUDic:al

ayltem

UIOCiatecl

wi1b

aodeI nllNIICe

adaptive

control, IEBB

TPrIIIf.

A.....,.

Coftt,••

vo l AC23.

pp. 499-5001'

Ja e 1971.

(4J K. S. NaNDdra uc l L S. Valavui, -Slable adaptive

COIltroUer

cIeIipa-Direct

coatrol.·,BU

TNar. Alii....,.

CfIIIIr.. vaL

ACn.

pp. S70-S83.

A

....

1911.

(5)

A. 'eur ad S. Mane,

..

Adapti¥c QOIltrol

of

. . . . . . . . . . .

t tiMar

. , . . . . . .

IEEB

n- t . A...... , .

C . .

.•

wi .

ACD,

pp.

557-570. A .... 1971.

(fi) G. A. DuIoDt u d P. L Bitaqer, --seu-hIIliq coatrol

oI.litaaiUDl dioxicte

kiIa.

IBEE

n.u. Aw

..... CMIr., 'VOl. A ~ Z

pp.

532-531, A -. 1971.

(7) K. J. Aatrim,

U.

IIoriIIoa,

L

liUDIt

aDdB. WitteDmUk,

-rbeory

ud

appticatiou

of

. .

t

replaton. A........, 19. pp. 457-476.

1977.

(I) L

l i

AuIyI i I 01 recuniw .toebutic

aJaoritJu.,

IEEE TIwu.

A

......

C. ,.., voL

Aen .

pp. '51-575. A - . 1m .

(9) - , OD

poei, ,

na l traDII.

hact io .

aDd

tIM

COIlwrpl lCe

of IOID8

recursive

,.. 1BEE

TIwu.

,A.,..,.

C.. . , wL ACn,

pp.

539-551, 1m.

(10) K. J.

A.aimud

B. Wkteamark. -oa leIf-tuaiAa replatan, vol. 9.

lIP-

195-199, 1973.

1'1) L lsillDl

ad

B.

WiUlllUDal'k. OD •

ltabiIiziDa

property

01 adaptive

replaton,

. , . , . I I 'AC

sy .

__ rtIMtl/blltJlt.

TbiIiIi.

u.s.s.R.,

1976.

(11] 8.

I pftb. -A

uaiIied approIdl to

aaodeI

rei.....adaptiYe

. , . . . . .

aIlcllelf

tuaiAa

replatan, Dep. Automat. CoDtr., LuaclIDat. TedmoL Tech. Rep., Dec. 1m .

[131

T.t

.........L V.

MoDopoIi,

DiIcrete

IDOcIeI

refereace adaptive

CODtrol

with aD

........

error1ipaI, AIIItIfIIIIIktJ.

vo l 13.

lIP-

507-517. Sept.

1m.

(t4)

J.

L

WiIkaI, 5MbIIUy,.... . ofD,,. . , .aI

S)..... New York: 1970.

(IS) G.

Co

000cIwiD. P. J. Ramadp.

uc l

P. Eo CaiDeI, MDiIcreCe time

.todautic

adaptive

ooatrol.

SIAM

J. c.u,. OptiMiz•• to be pubIiIbecL

(lfi) A. S.

Mone,

-cHoba1atability of

parameter

adaptive

OODtrol.,.

..... Yale UDiv.•

S A IS Rep.

1I02L Mar.

1979.

(17]

x.. S. Nueadra

aDd

V-H.

LiB.

-Stable

ctilcrete adaptive ccmtrol, Yale UDiv., S •

IS

Rep. 79011' Mar.

1979.

[II)

B.1 pftb.

-Stability of modcI refcreace adaptive and

IeJf

tUDiq rqulaton, Cep.

Automat.

Caatr

Of

Lad

last.

TecbAo1.,

TedL

Rep., Dec. 1978.

494

discrete-time multivariable adaptive control

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